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##############################################################################
##
#W lcset.gi GAP4 package `Utils' Chris Wensley
##
#Y Copyright (C) 2015-2025, The GAP Group
#############################################################################
##
#O LeftCoset( <g>, <U> ) . . . . . . . . forms a left coset of U by g \in G
##
InstallMethod( LeftCoset, "generic method for left cosets", true,
[ IsObject, IsGroup ], 0,
function( g, U )
local lc, fam;
fam := FamilyObj( U );
if not IsBound( fam!.leftCosetsDefaultType ) then
fam!.leftCosetsDefaultType :=
NewType( fam, IsLeftCosetDefaultRep and HasActingDomain
and HasFunctionAction and HasRepresentative );
fi;
lc := rec();
ObjectifyWithAttributes( lc, fam!.leftCosetsDefaultType,
Representative, g,
ActingDomain, U,
FunctionAction, OnRight,
IsLeftCoset, true );
if HasSize( U ) then
SetSize( lc, Size( U ) );
fi;
return lc;
end);
#############################################################################
##
#A Inverse( <lc> ) . . . . . forms the right coset Ug^-1 corresponding to gU
#A Inverse( <rc> ) . . . . . forms the left coset g^-1U corresponding to Ug
##
InstallOtherMethod( Inverse, "generic method for left cosets", true,
[ IsLeftCoset ], 0,
function( lc )
local g, U, rc;
g := Representative( lc );
U := ActingDomain( lc );
rc := RightCoset( U, g^-1 );
SetInverse( rc, lc );
return rc;
end);
InstallOtherMethod( Inverse, "generic method for right cosets", true,
[ IsRightCoset ], 0,
function( rc )
local g, U, lc;
g := Representative( rc );
U := ActingDomain( rc );
lc := LeftCoset( g^-1, U );
SetInverse( lc, rc );
return lc;
end);
InstallMethod( PrintString, "for a left coset", true, [ IsLeftCoset ], 0,
function( d )
return STRINGIFY( "LeftCoset(\<",
PrintString( Representative(d) ), ",\>",
PrintString( ActingDomain(d) ), ")" );
end);
InstallMethod( PrintObj, "for a left coset", true, [ IsLeftCoset ], 0,
function( d )
Print( PrintString( d ) );
end);
InstallMethod( ViewString, "for a left coset", true, [ IsLeftCoset ], 0,
function( d )
return STRINGIFY( "LeftCoset(\<",
ViewString( Representative(d) ), ",\>",
ViewString( ActingDomain(d) ), ")" );
end);
InstallMethod( ViewObj, "for a left coset", true, [ IsLeftCoset ], 0,
function( d )
Print( ViewString( d ) );
end);
InstallMethod( \=, "for two left cosets", IsIdenticalObj,
[ IsLeftCoset, IsLeftCoset ], 0,
function( lc1, lc2 )
return ( ActingDomain( lc1 ) = ActingDomain( lc2 ) ) and
( Representative(lc1)/Representative(lc2) in ActingDomain( lc1 ) );
end);
InstallMethod( \in, "element and LeftCoset", true,
[ IsMultiplicativeElementWithInverse, IsLeftCoset ], 100,
function( g, lc )
return ( Representative(lc)^-1 * g ) in ActingDomain( lc );
end);
InstallOtherMethod( \*, "element and LeftCoset", true,
[ IsMultiplicativeElementWithInverse, IsLeftCoset ], 0,
function( g, lc )
return LeftCoset( g * Representative( lc ), ActingDomain( lc ) );
end);
InstallOtherMethod( \^, "LeftCoset and element", true,
[ IsLeftCoset, IsMultiplicativeElementWithInverse ], 0,
function( lc, g )
return g^-1 * lc;
end);
InstallMethod( Size, "for a left coset", true, [ IsLeftCoset ], 0,
function( lc )
return Size( ActingDomain( lc ) );
end);
InstallOtherMethod( IsBiCoset, "for a left coset", true, [ IsLeftCoset ], 0,
function( lc )
return IsBiCoset( Inverse( lc ) );
end);
InstallOtherMethod( AsSet, "for a left coset", true, [ IsLeftCoset ], 0,
function( lc )
local L;
L := AsSet( Inverse( lc ) );
return Set( L, g -> g^-1 );
end);
InstallOtherMethod( Intersection2, "for two left cosets", true,
[ IsLeftCoset, IsLeftCoset ], 0,
function( lc1, lc2 )
local L;
L := Intersection2( Inverse( lc1 ), Inverse( lc2 ) );
return Set( L, g -> g^-1 );
## return Inverse( L ); to be used once 4r12 is out
end);
#############################################################################
##
#F LeftCosets( <G> <U> ) . . . . . . . . . . . . left cosets of U by g \in G
#O LeftCosetsNC( <G> <U> ) . . . . . . . . . . . left cosets of U by g \in G
##
BindGlobal( "LeftCosets",
function( G, U )
if not IsSubset( G, U ) then
Error("not contained");
fi;
return LeftCosetsNC( G, U );
end );
InstallMethod( LeftCosetsNC, "generic", IsIdenticalObj, [ IsGroup, IsGroup ], 0,
function( G, U )
local L;
L := RightCosetsNC( G, U );
return List( L, Inverse );
end);
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